Question

lines AC and BD are diagonals of ABCD. What value of x makes ABCD a rectangle if AC=x+7 and BD=3x-5

Answer 1

For the quadrilateral \( ABCD \) to be a rectangle, the diagonals \( AC \) and \( BD \) must be equal in length.

Given:
- \( AC = x + 7 \)
- \( BD = 3x - 5 \)

To find the value of \( x \) that makes \( ABCD \) a rectangle, we set the two expressions for the diagonals equal to each other:

\[
x + 7 = 3x - 5
\]

Now, let's solve for \( x \):

1. Subtract \( x \) from both sides:

\[
7 = 2x - 5
\]

2. Add 5 to both sides:

\[
12 = 2x
\]

3. Divide both sides by 2:

\[
x = 6
\]

Now, we can check whether this value makes the lengths of the diagonals equal:

- Calculate \( AC \):

\[
AC = x + 7 = 6 + 7 = 13
\]

- Calculate \( BD \):

\[
BD = 3x - 5 = 3(6) - 5 = 18 - 5 = 13
\]

Since both \( AC \) and \( BD \) equal 13 when \( x = 6 \), this value confirms that quadrilateral \( ABCD \) is a rectangle.

Thus, the value of \( x \) that makes \( ABCD \) a rectangle is:

\[
\boxed{6}
\]